Advertisement

Exact instanton expansion of superconformal Chern-Simons theories from topological strings

  • Sanefumi Moriyama
  • Tomoki NosakaEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the CalabiYau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the same expression from the refined topological string theory as the ABJM matrix model with different topological invariants while that of the other is more general. We also observe an unexpected relation between these two theories.

Keywords

Matrix Models Chern-Simons Theories Topological Strings M-Theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  2. [2]
    O. Aharony, O. Bergman, D.L. Jafferis and J.M. Maldacena, \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Mariño and P. Putrov, Exact Results in ABJM Theory from Topological Strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    H. Fuji, S. Hirano and S. Moriyama, Summing Up All Genus Free Energy of ABJM Matrix Model, JHEP 08 (2011) 001 [arXiv:1106.4631] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Mariño and P. Putrov, ABJM theory as a Fermi gas, J. Stat. Mech. 1203 (2012) P03001 [arXiv:1110.4066] [INSPIRE].MathSciNetGoogle Scholar
  9. [9]
    M. Hanada, M. Honda, Y. Honma, J. Nishimura, S. Shiba and Y. Yoshida, Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant, JHEP 05 (2012) 121 [arXiv:1202.5300] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact Results on the ABJM Fermi Gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    P. Putrov and M. Yamazaki, Exact ABJM Partition Function from TBA, Mod. Phys. Lett. A 27 (2012) 1250200 [arXiv:1207.5066] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton Effects in ABJM Theory from Fermi Gas Approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    F. Calvo and M. Mariño, Membrane instantons from a semiclassical TBA, JHEP 05 (2013) 006 [arXiv:1212.5118] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton Bound States in ABJM Theory, JHEP 05 (2013) 054 [arXiv:1301.5184] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
  19. [19]
    M.-x. Huang and A. Klemm, Direct integration for general Ω backgrounds, Adv. Theor. Math. Phys. 16 (2012) 805 [arXiv:1009.1126] [INSPIRE].
  20. [20]
    J. Choi, S. Katz and A. Klemm, The refined BPS index from stable pair invariants, Commun. Math. Phys. 328 (2014) 903 [arXiv:1210.4403] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-Matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    D.L. Jafferis and A. Tomasiello, A Simple class of \( \mathcal{N}=3 \) gauge/gravity duals, JHEP 10 (2008) 101 [arXiv:0808.0864] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    H.-C. Kao and K.-M. Lee, Selfdual Chern-Simons systems with an N = 3 extended supersymmetry, Phys. Rev. D 46 (1992) 4691 [hep-th/9205115] [INSPIRE].ADSGoogle Scholar
  24. [24]
    H.-C. Kao, K.-M. Lee and T. Lee, The Chern-Simons coefficient in supersymmetric Yang-Mills Chern-Simons theories, Phys. Lett. B 373 (1996) 94 [hep-th/9506170] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    Y. Imamura and K. Kimura, \( \mathcal{N}=4 \) Chern-Simons theories with auxiliary vector multiplets, JHEP 10 (2008) 040 [arXiv:0807.2144] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    S. Moriyama and T. Nosaka, Partition Functions of Superconformal Chern-Simons Theories from Fermi Gas Approach, JHEP 11 (2014) 164 [arXiv:1407.4268] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    C.P. Herzog, I.R. Klebanov, S.S. Pufu and T. Tesileanu, Multi-Matrix Models and Tri-Sasaki Einstein Spaces, Phys. Rev. D 83 (2011) 046001 [arXiv:1011.5487] [INSPIRE].ADSGoogle Scholar
  28. [28]
    D.R. Gulotta, C.P. Herzog and S.S. Pufu, From Necklace Quivers to the F-theorem, Operator Counting and T(U(N)), JHEP 12 (2011) 077 [arXiv:1105.2817] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S. Moriyama and T. Nosaka, ABJM Membrane Instanton from Pole Cancellation Mechanism, arXiv:1410.4918 [INSPIRE].
  30. [30]
    A. Grassi and M. Mariño, M-theoretic matrix models, JHEP 02 (2015) 115 [arXiv:1403.4276] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    Y. Hatsuda and K. Okuyama, Probing non-perturbative effects in M-theory, JHEP 10 (2014) 158 [arXiv:1407.3786] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    M.B. Green, J.H. Schwarz and L. Brink, Superfield Theory of Type II Superstrings, Nucl. Phys. B 219 (1983) 437 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    C.A. Tracy and H. Widom, Proofs of two conjectures related to the thermodynamic Bethe ansatz, Commun. Math. Phys. 179 (1996) 667 [solv-int/9509003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    M.-x. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  36. [36]
    Y. Hatsuda, M. Honda and K. Okuyama, work in progress.Google Scholar
  37. [37]
    H. Awata, S. Hirano and M. Shigemori, The Partition Function of ABJ Theory, Prog. Theor. Exp. Phys. (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
  38. [38]
    M. Honda, Direct derivation ofmirrorABJ partition function, JHEP 12 (2013) 046 [arXiv:1310.3126] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    M. Honda and K. Okuyama, Exact results on ABJ theory and the refined topological string, JHEP 08 (2014) 148 [arXiv:1405.3653] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    S. Matsumoto and S. Moriyama, ABJ Fractional Brane from ABJM Wilson Loop, JHEP 03 (2014) 079 [arXiv:1310.8051] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, \( \mathcal{N}=4 \) Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets, JHEP 07 (2008) 091 [arXiv:0805.3662] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    M. Honda and S. Moriyama, Instanton Effects in Orbifold ABJM Theory, JHEP 08 (2014) 091 [arXiv:1404.0676] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    D. Gaiotto and E. Witten, Janus Configurations, Chern-Simons Couplings, And The θ-Angle in \( \mathcal{N}=4 \) Super Yang-Mills Theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Kobayashi Maskawa Institute & Graduate School of MathematicsNagoya UniversityNagoyaJapan
  2. 2.Yukawa Institute for Theoretical PhysicsKyoto UniversitySakyo-kuJapan

Personalised recommendations